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2 Quadratic Equations in One Unknown (II)

Review Exercise 2 (p. 2.5)

∴ (repeated)

4. Using the quadratic formula,

∵ is not a real number.∴ The equation has no real roots.

7. The x-intercepts of the graph of are –2.0 and 3.0.Therefore, the roots of are –2.0 and 3.0.

NSS Mathematics in Action (2nd Edition) 4A Full Solutions

8. The x-intercept of the graph of is 0.5.

Therefore, the root of is 0.5.

9. (a) The graph of intersects the x-axis at two points.

Therefore, the equation has two unequal real roots.

(b) The graph of touches the x-axis at one point.Therefore, the equation has one double real root.

(c) The graph of does not intersect the x-axis.

Therefore, the equation has no real roots.

To Learn More

To Learn More (p. 2.23)

(a) Mid-point of PQ

(b) Length of PQ

To Learn More (p. 2.30)(a)

Classwork

Classwork (p. 2.7)

Value of ( = b2

– 4ac)

Nature of roots

2 unequal real roots

1 doublereal root

No realroots

(a) 0 (b) 24 (c) – 31 (d) – 400 (e) 25

Classwork (p. 2.10)1. (a) 0 (b) < 0

2. (a) 2 (b) > 0

3. (a) 0 (b) < 0

4. (a) 1 (b) = 0

5. (a) 1 (b) = 0

6. (a) 2 (b) > 0

Classwork (p. 2.18)

Sum of roots Product of roots

Classwork (p. 2.29)1. (a)

2. (a)

Classwork (p. 2.31)

Real part Imaginary part

(a) 3 – 4

(b) – 5 7

(c) 8 8 0

Classwork (p. 2.32)(a) (b) (c) (d)

Quick Practice

Quick Practice 2.1 (p. 2.7)∵ The equation has one double real root.∴ = 0

Quick Practice 2.2 (p. 2.8)∵ The equation has no real roots.∴

∴ The range of values of k is .

Quick Practice 2.3 (p. 2.9)(a) For the equation ,

(b) ∵ is a quadratic equation.

∴ The coefficient of cannot be zero.

(i) ∵ The equation has two distinct real roots.

∴ The range of values of k is except .

(ii) ∵ The equation has real roots.

∴ The range of values of k is except .

Quick Practice 2.4 (p. 2.11)(a) ∵ The graph of touches the x-axis at one point P.

(b) For m = 9, the corresponding quadratic equation is

∴ The coordinates of P are .

Quick Practice 2.5 (p. 2.11)(a) ∵ The graph of has two

x-intercepts.∴

∴ The range of values of m is .

(b) The smallest integral value of m is 0.For m = 0, the corresponding quadratic equation is

∴ The x-intercepts of the graph are and 1.

Quick Practice 2.6 (p. 2.15)(a) The required quadratic equation is:

(b) The required quadratic equation is

Quick Practice 2.7 (p. 2.16)

(a) Sum of roots

Product of roots

∴ The required quadratic equation is

(b) Sum of roots

Product of roots

Quick Practice 2.8 (p. 2.19)(a) For the equation ,

Quick Practice 2.12 (p. 2.23)∵ and are the roots of .

For the required quadratic equation,

By comparing the real parts, we have

By comparing the imaginary parts, we have

Quick Practice 2.18 (p. 2.36)Using the quadratic formula,

Quick Practice 2.19 (p. 2.37)∵ is a root of the equation .

∴From (2),

By substituting into (1), we have

Further Practice

Further Practice (p. 2.12)

∵ The equation has two unequal real roots.∴

∴ The range of values of m is m < 8.

2. (a) ∵ The graph of touches the x-axis at one point P.

(b) For k = 5, the corresponding quadratic equation is

For , the corresponding quadratic equation is

2. (a)

By comparing the imaginary parts, we have……(1)

Exercise

Exercise 2A (p. 2.12)Level 11. For the equation ,

∵∴ The equation has two unequal real roots.

2. For the equation ,

∵∴ The equation has no real roots.

∵∴ The equation has one double real root.

4. Consider .

∴ The graph of has one x-intercept.

5. Consider .

∴ The graph of has no x-intercepts.

6. Consider .

∴ The graph of has two x-intercepts.

7. ∵ The equation has one double real root.

8. ∵ The equation has two equal real

roots.∴

9. ∵ The equation has two unequal real roots.

10. ∵ The equation has two unequal real roots.

11. ∵ The equation has no real roots.∴

12. ∵ The equation has no real roots.

∴ The range of values of k is k > 5.

13. ∵ The graph of has only one x-intercept.

14. ∵ The graph of has no x-intercepts.∴

15. ∵ The equation has real roots.

17. ∵ The graph of touches thex-axis at only one point.

18. ∵ The graph of cuts the x-axis at two distinct points.

Level 2

∵ The equation has one double real root.

∵ The equation has two distinct real roots.

∵ is a quadratic equation.∴ The coefficient of x2 cannot be zero.i.e.

∴ The range of values of m is except .

21. (a) ∵ The graph of touches the x-

axis at one point P.∴

∴ The coordinates of P are (–6, 0).

22. (a)

∵ The graph of touches the x-axis at one point Q.

(b) For , the corresponding quadratic equation is

∴ The coordinates of Q are .∴ Length of OQ

∵ The graph of has no x-intercepts.

∴ The range of values of p is .

∵ The equation has real roots.∴

∴ The largest value of k is .

∵ The graph of intersects the x-axis.

∴ The smallest value of m is .

26. (a)

∵ The graph of cuts the x-axis at two points.

(b) The largest integral value of p is 0.For p = 0, the corresponding quadratic equation is

∴ The x-intercepts of the graph are and 1.

27. (a)

∵ The equation has two unequal real roots.

(b) The smallest integral value of k is .For , the corresponding quadratic equation is

∴ The equation has two unequal real roots for any positive values of p.

29. ∵ The equation has two distinct real roots.

∴ Any pair of integral values of a and c such that

is acceptable.

(or any other reasonable answers)

30. ∵ The graph of touches the x-axis at one point.

∴ Any pair of integral values of m and n such that

m2 = 4n is acceptable.

(or any other reasonable answers)

31. (a)

(i) ∵ The equation has

two distinct real roots.∴

∴ The range of possible values of k is

(ii) The only possible negative integral value of k is .For , the corresponding quadratic

equation is

Exercise 2B (p. 2.17)Level 11. The required quadratic equation is

2. The required quadratic equation is

Level 2

12. (a)

(b) The roots of the required quadratic equation areand , i.e. 0 and 6 respectively.

13. (a)

(b) The roots of the required quadratic equation are

and , i.e. and respectively.

14. (a)

and , i.e. and respectively.

15. (a) Using the quadratic formula,

, i.e. .

16. (a)

Using the quadratic formula,

, i.e. .

17. (a) The required quadratic equation is

(b) When m = 1, n = 6 or m = 6, n = 1.The required quadratic equation is

When m = 2, n = 3 or m = 3, n = 2.The required quadratic equation is

Exercise 2C (p. 2.24)Level 1

5. For the equation mx2 + 9x + n = 0,

6. Let be the other root.

∴ The other root is 5.

7. (a)

8. (a)

When ,

14. ∵ and are the roots of x2 – 5x – 3 = 0.

(a) For the required quadratic equation,

(b) For the required quadratic equation,

15. ∵ and are the roots of .

(a) For the required quadratic equation,

(b) For the required quadratic equation,

16. (a) Sum of roots =

Product of roots =

(b) ∵ –2 and 3 are the x-intercepts of the graph of y = px2 + qx – 12.

∴ –2 and 3 are the roots of px2 + qx – 12 = 0.

17. (a) (i) By substituting (0, 15) into y = –x2 + mx + n,we have

(ii) ∵ 5 is one of the x-intercepts of the graph ofy = –x2 + mx + n.

∴ 5 is one of the roots of –x2 + mx + n = 0.Let be the other root.

∴ The coordinates of P are (–3, 0).

Level 2

20. (a)

∵ Sum of roots = product of roots + 6

(b) By substituting k = 5 into the equation, we have

21. Let and be the roots of the equation.

22. Let and be the roots of , where.

∵ The difference between the roots is 3.

26. Sum of roots =

Product of roots =

(c) From (b), (rejected ∵ )

(d) From (c), (rejected∵ )

32. (a) (i) ∵ is a root of .

(b) ∵ is also a root of .

33. Let and be the roots of rx2 – (r + 2)x + 2 = 0, where < .∴ The coordinates of A and B are (, 0) and (, 0)

respectively.

(a) ∵

(b) By substituting into the equation, we have

∴ The coordinates of A and B are (1, 0) and (4, 0) respectively.

34. Let and be the roots of , where < .∴ The coordinates of A and B are (, 0) and (, 0)

respectively.

(a) ∵ P(5, 0) is the mid-point of AB.

∴ The coordinates of A and B are (2, 0) and (8, 0) respectively.

∴ The length of AB

Exercise 2D (p. 2.37)Level 1

1. (a)

2. (a)By comparing the real parts, we have

Level 227.

∴From (1),

∴From (2),

35. (a)

36. ∵ is a root of the equation .

∴From (2),

37. ∵ is a root of the equation .

∴From (2),

38. (a)

∵ is a root of the equation

∴From (2),

39. (a) ∵ is a root of the equation .

∴If , then

∴ is also a root of .∴ Sharon’s claim is agreed.

Check Yourself (p. 2.41)1. (a) (b) (c)

(d) (e)

3. ∵ The graph of has only one x-intercept.

4. ∵ The equation has no real roots.∴

5. (a)

7. (a)By comparing the real parts, we have

8. (a)

Revision Exercise 2 (p. 2.42)Level 11. For the equation ,

∴ The equation has two unequal real roots and the corresponding graph has two x-intercepts.

∴ The equation has one double real root and the corresponding graph has one x-intercept.

∴ The equation has no real roots and the corresponding graph has no x-intercepts.

∴ The equation has two unequal real roots and the

corresponding graph has two x-intercepts.

5. ∵ The graph of has two x-intercepts.

∴ The range of values of p is p < 1.

6. ∵ The graph of does not intersect the x-axis.

(a) ∵ The equation has two unequal real roots.

∴ The range of values of k is k < 6.

(b) ∵ The equation has no real roots.

∴ The range of values of k is k > 6.

9. (a) ∵ The graph of has only one

x-intercept.∴

∴ The x-intercept of the graph is .

10. (a)

∵ The equation has two equal real roots.

14. (a)

16. Let and be the roots of .

When ,When ,∴

(c) The required quadratic equation is

24. Let and be the roots of .

25. ∵ The graph of cuts the x-axis at

∴ The roots of are and .

26. (a) (i) ∵ The graph oftouches the x-axis at one point.

(ii) By substituting into, we have

∵ The graph of cuts the y-axis at Q.

∴ By substituting into , we have

∴ The coordinates of Q are .

27. (a)

28. (a)

29. (a)

∴From (2),

30. (a)

(b) Using the quadratic formula,

(c) Using the quadratic formula,

Level 2

31. (a)

(b) If k is a negative integer, then , i.e. the equationhas no real roots.∴ Peter’s claim is agreed.

32. (a) ∵ The equation has one double real root.

(b) By substituting into , we have

∴ The x-intercept of the graph is –2.

33. (a) ∵ The graph of

cuts the x-axis at two points.∴ is a

quadratic equation.∴ The coefficient of cannot be zero.

(b) ∵ The graph of

cuts the x-axis at two points.∴

∴ The range of values of m is except

34. (a) ∵ The graph of y = ax2 + 8x + c touches the x-axis at only one point.

∴i.e.

Since a and c are positive integers,when a = 1 and c = 16, a + c = 1 + 16 = 17;when a = 2 and c = 8, a + c = 2 + 8 = 10;when a = 4 and c = 4, a + c = 4 + 4 = 8;when a = 8 and c = 2, a + c = 8 + 2 = 10;when a = 16 and c = 1, a + c = 16 + 1 = 17.∴ The possible values of a + c are 8, 10 and 17.

(b) For the equation 2x2 + (a + c)x + 10 = 0,Δ = (a + c)2 – 4(2)(10) = (a + c)2 – 80When a + c = 8,Δ = 82 – 80 = –16 < 0In this case, the equation 2x2 + (a + c)x + 10 = 0 has no real roots.∴ Ken’s claim is not correct.

For the equation ,

∴ The quadratic equationhas two distinct real roots for any real values of k.

36. (a) ∵ is a quadratic equation.

∵ Sum of roots is equal to product of roots.

(b) If the sum of its roots is equal to the product of its roots, then .By substituting into

, we have

∴ The roots of the quadratic equation must be distinct.

∴ Kelvin’s claim is agreed.

(∵ )

For the quadratic equation ,

42. (a)

43. ∵ 2 and 2 are the roots of .

44. ∵ The graph of intersects the x-axis at two points.

∴ The equation has two distinct real roots.

Also, the y-intercept of the graph is positive.

∴ The possible integral values of k are 1, 2 and 3.

45. (a) ∵ The graph of does not intersect the x-axis.∴

(b) When x = 1,

For , .

∴ It is possible that lies on the graph.

46. Consider the quadratic equation.

∴ The quadratic equation has two distinct real roots.∴ The graph of has

two x-intercepts.∴ The graph is not correctly sketched.

47. (a) Let and be the roots of ,where .

∵∴

(b) x-coordinate of M

∴ The coordinates of M are .

48. (a)

49. (a)

50. (a)

∴From (1),

∴(1) + (2):

51. (a)

∵ The imaginary part of z is 2.

(b) By substituting into (1), we have

52. Let , where a and b are real numbers.

∵ is an imaginary number.

∵ is a real number.

(a) ∵ The equation has two non-real roots.∴

∴ The range of values of m is m > 6.

(b) The smallest integral value of m is 7.By substituting m = 7 into , we have

Using the quadratic formula,

54. (a) ∵ is a root of the equation.

∴From (2),

(b) By substituting and into

, we have

∵ The equation has real roots.∴

55. (a)

56. (a)

Multiple Choice Questions (p. 2.47)1. Answer: C

∴ Only II and III have real roots.∴ The answer is C.

2. Answer: D∵ The equation has no real roots.

∴i.e.

∴ The range of values of p is p > 16.3. Answer: D

∵ The equation has one double real root.

4. Answer: A∵ The graph of cuts the x-axis at two

points.∴i.e.

∴ The possible values of c are 15 and 20.

5. Answer: D

6. Answer: D∵∴ a and b are the roots of .

7. Answer: D∵ is a root of .

∵ and are the roots of .

8. Answer: C

The required quadratic equation is

9. Answer: B

10. Answer: A

∴ The real part is .

11. Answer: D

12. Answer: BLet , where a and b are real numbers.

By substituting a = 3 into (1), we have

HKMO (p. 2.48)1. ∵ 1 is a root of .

∵ a and b are roots of .

2. Consider .

Consider .

∴ The roots of are 2006 and 2007.

Exam Focus

Exam-type Questions (p. 2.50)1. (a) ∵ The graph of cuts the y-axis at

C(0, –5).∴ By substituting x = 0 and y = –5 into

, we have

(b) ∵ and are the x-intercepts of the graph of .

∴ and are the roots of .

2. (a) ∵ The graph of cuts the

y-axis at C(0, –5).∴ By substituting x = 0 and y = –5 into

, we have

(b) ∵ and are the x-intercepts of the graph of .

∴ and are the roots of .

(c) Coordinates of M

Distance between M and C

∴ The distance between M and C is not greater than 6.

3. (a)

(b) (i) ∵ is a root of the equation

(ii) When , the quadratic equation becomes

∵ The equationhas two

distinct real roots.∴

∴ The range of values of r is .

4. Answer: A

∵ The equation has a repeated root.

5. Answer: A

6. Answer: A

∴ Only I and II are real.∴ The answer is A.

Investigation Corner (p. 2.53)

2. Let x and y be the length and the width of the rectangle respectively.From the question, we have and .Solution Steps done by

BabyloniansAlgebraic Notations added to facilitate understanding

, denoted as P

, denoted as Q2

, which equals x

, which equals y

∴ The dimensions of the rectangle are (or).

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